HA NOI UNIVERSITY OF MINING AND GEOLOGY
TEST 1
Total time: 90 minutes
1. Given that
lim
𝑥→𝑐
𝑓
𝑥=𝐿 and
lim
𝑥→𝑐
𝑔
𝑥=𝑀 and that f(x) ≤ g(x) for all x in an open interval containing c (except possibly c itself). Prove that: L ≤ M.
2. Find the limits: 1)
2)

3. Find the limit:
lim
𝑥→1
2𝑥(𝑥−1|𝑥−1

4. For what values of a and b is 𝑓
𝑥
𝑎𝑥+2𝑏 𝑥≤0
𝑥
2+3𝑎−𝑏 0<𝑥≤2
3𝑥−5 𝑥>2 continuous at every x?
5*.Prove that if f(x) is a function defined on R and sastified: 𝑓
𝑥+1+𝑓
𝑥−1
2
𝑓(𝑥) then f(x) is a periodic function and find its period?








HA NOI UNIVERSITY OF MINING AND GEOLOGY
TEST 1
Total time: 90 minutes

1. Given that
lim
𝑥→𝑐
𝑓
𝑥=𝐿
and
lim
𝑥→𝑐
𝑔
𝑥=𝑀. Prove that:
lim
𝑥→𝑐
𝑓
𝑥+𝑔
𝑥=𝐿+𝑀.
2. Find the limits: 1)
2)

3. Find the limit:
lim
𝑥→−2(𝑥+3|𝑥+2
𝑥+2

4. For what values of a and b is 𝑓
𝑥−2 𝑥≤−1
𝑎𝑥−𝑏 −1<𝑥<1
3 𝑥≥1 continuous at every x?
5*. Prove that if f(x) is a function defined on R and sastified: 𝑓
𝑥+1+𝑓
𝑥−1
2
𝑓(𝑥) then f(x) is a periodic function and find its period?
HA NOI UNIVERSITY OF MINING AND GEOLOGY
FINAL TEST
Total time: 90 minutes
1. Prove that: If f has a local maximum or minimum value at an interior point c of its domain, and if f’ is defined at c, then f’(c) = 0.
2. Find the limits:
1)
lim
𝑥→0(𝑥
𝑒
2𝑥
1
𝑥
2)
lim
𝑥
𝜋
3
sin⁡(𝑥
𝜋
3
1−2𝑐𝑜𝑠𝑥

3. Find f(0) to make f(x) continuous at x = 0 if
f(x) =
𝑠𝑖𝑛𝑎𝑥−𝑠𝑖𝑛𝑏𝑥
𝑥

4. Find f(2013)(0) if f(x) =
1
𝑥
2−3𝑥+2.
5. Write formular for fog and find the domain and range if
f(x) = 2 – x2 , g(x) =
2+𝑥

6. Let f: (0; () ( R be a twice continuously differentiable function such that
|f’’(x) + 2xf’(x) + (x2 + 1)f(x)| ≤ 1
For all x. Prove that
lim
𝑥
𝑓
𝑥=0.





HA NOI UNIVERSITY OF MINING AND GEOLOGY
FINAL TEST
Total time: 90 minutes
1. Prove that: Suppose that y = f(x) is continuos at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b). If f(a) = f(b) then there is at least one number c in (a, b) at which f’(c) = 0.
2. Find the limits:
1)
lim
𝑥→1(1−𝑥
𝑐𝑜𝑠
2
𝑥
2)
lim
𝑥→0
𝑎𝑟𝑐𝑡𝑎𝑛𝑥
𝑠𝑖𝑛𝑥−𝑥

3. Find f(0) to make f(x) continuous at x = 0 if
f(x) =
𝑒
𝑎𝑥
𝑒
𝑏𝑥
𝑥

4. Find f(7)(x) if f(x) =
𝑥
3
1−𝑥.
5. Write formular for gof and find the domain and range if
f(x) = 2 – x2 , g(x) =
2+𝑥

6. Prove that if f: R (R is three times differentiable, then there exists a real number (( (-1, 1) such that

𝑓(𝜖
6
𝑓
1−𝑓(−1
2
𝑓(0
HA NOI UNIVERSITY OF MINING AND GEOLOGY
FINAL TEST
Total time